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I have seen several attempts at mathhammer and most of them use law of averages which is near useless. I start this tactica by saying I am a business/finance major, this has little significance but because of this I have taken courses such as statistics and regression analysis. In these classes you are taught how people/companies use standard deviation to make accurate probability distrubitions. I have decided to try to use these principles to create real mathhammer. When people refer to mathhammer they usually think of averages, but this is not the case in this tactica I am using standard deviation to show how a player can predict what he/she will roll 68% of the time.(Yes, thats right 68% guranteed)
I am a Imperial Guard player so I will use our most common weapon the lasgun for most of my examples.
First using averages for mathhammer is ok but the player will only be right 50% of the time. I will use an example to show this.
Example
If I rapid fire 10 lasguns with a bs of 3 the average number of hits is 10. According to the averages there will be 5 force saves against toughness of 3.
Yes, this is true but it only happens 50% of the time. In a game that last five turns I do not believe it is beneficial for a player to base a decision in one of his/her five shooting phases on a statistic that is essentially a flip of a coin. By knowing and using standard deviation a player will have a realistic range of the probabilities making his/her decision process easier.
Standard deviation is the distance away from the mean or average. One standard deviation away from the mean happens 68% of the time two standard deviations away from the mean happens 85% of the time, 3 standard deviations away happens 99.7% of the time this is a statistical fact, and it is called the 68-95-97 rule.(it is neat stuff the reader can google it to find out more)
I recommend all readers get out their dice and start rolling while reading the tactica so that one can see that it works for his/herself. Now to make this rule helpful, I will apply it to 10 lasgun shots with a bs of 3.
10 shots average number of hits is 5
5 hits is the average or mean in our example.
Next we go one standard deviation away from the mean by adding one or subtracting one.
5-1=4
5+1=6
4 and 6 is one standard deviation away
This means that 68% of the time a player firing 10 lasgun shots with bs 3 will have either 4 hits, 5 hits, or 6 hits. To test this get out 10 dice then roll them 10 seperate times, keep up with the number of times you actually roll 4 hits, 5 hits, or 6 hits out of ten. How many times did you roll 4/5/6 hits, I believe it was six or seven times out of ten. After doing the exercise the reader will come to the conclusion that by using standard deviation he/she can come up with a probability distribution that is right 68% of the time, know we have numbers on our side we have real mathhammer.
Now using real mathhammer lets use some examples to create an accurate probability distributition range that is helpful on the table top. To keep things simple we will use a toughness of 3 in our examples.
We know that 68% of the time we will roll 4/5/6 hits. So how does a player use this stat to decide how many hits he will roll? The player decideds that, but if he/she uses the lowest number you will know the minimum amount of damage you can statistcally expect, and if you use the highest you know the maximum amount of damage you can expect.
Example
4 lasgun hits averages 2 force saves against toughness 3.
Apply standard deviation
2-1=1
2+1=3
If you roll 4 hits you can expect 1-3 saves 68% of the time.
5 lasgun hits averages 2.5 force saves against toughness 3
Apply Standard deviation
2.5-1=1.5
2.5+1=3.5
If you roll 5 hits you can expect to force between 1.5-3.5 saves 68% of the time, or 1-3 if you want to know the minimum or 2-4 maximum estimate.
6 lasgun hits averages 3 force saves against toughness 3
Apply standard deviation
3-1=2
3+1=4
6 lasgun hits forces 2-4 saves 68% of the time.
Now we put all of it together 10 lasgun shots with a bs of 3 against a toughness 3 unit will 68% of the time force 1-4 saves.
With all that said it only works 68% of the time however it is statistically proven to work 68% of the time. In a game that is base off of statistics to determine point value, knowing real mathhammer can be used to determine what a unit will do 68% of the time.
I hope this helps
thanks
Nice work. This is useful as the range of results is more likely than the mean average.
Readers should note, however, that it is all still averages based on dice rolls which are not purely random. No matter how you roll dice, their results are still dependent on outside factors such as what their original orientation was, how they were rolled, and what they were rolled onto.
For example, my test rolls for the standard deviation yielded 4, 5, or 6 hits only 5/10 times. I actually got 7-9 hits 3/10 times, which is abnormal spread if you go by averages or standard deviation. It could be a result of random variations, but I normally roll that well during the course of several games while others may not. This to me means that there is something to our "gut feelings" as we roll the dice. These feelings are probably our mind abstracting the many variables the we could not analyze individually and giving us a "good" or "bad" probability through our emotional responses.
The lesson here is: don't blindly believe in averages if you aren't feeling good about the roll. Change something up first before dropping the dice down.
I preffer my method (though yours is undoubtably useful). I predict the probability I will achieve the outcome I want rather then work out the outcomes I am most likely to achieve.
For instance, I want to stop a unit of 5 kroot (origional size 10) capturing an objective. It is currently my movement phase in turn 6 and my opponent does not have another turn. I have a unit of 10 Guardsmen with lasguns (further then 12" away) that can only see the kroot. I want to know the probability that the kroot will not claim the objective without having to shoot any unit other then the 10 guardsmen. To remove the kroot I either need to:
Kill all 5 (7.18%)
Kill 2-4 (67.79%) and the Kroot fail their LD 6 test (58.33%) (34.54% of both occuring)
So my total chance of removing the Kroot with the guardsmen is 46.7%. With that percent I can make decisions on wether I should move other units to get shots on the Kroot, or wether they should go for other objectives.
Last edited by kroxigor01; January 27th, 2009 at 08:37.
Ckok, as someone with a cursory understanding of the subject at hand, I applaud the effort to move people beyond basic probabilities and into the realm of higher math, but I think there are a few confusing elements to your treatment.
First, how is the standard deviation determined for any given range of outcomes? Doesn't one have to know the probabilities anyway in order to find the standard deviation?
Second, what does it tell us that probabilities don't? In your example of rolling to hit with ten lasgun shots, I could look at the three most likely outcomes (5, 4 and 6) and find the probability of getting one of the three, and the result would be more accurate and faster than the standard deviation you give.
This would look like P(A or B or C) = P(A) + P(B) + P(C) = 0.246 + .205 + .205 = .656, meaning that I've got a 65.6% chance of rolling four, five or six hits.
Perhaps I've never seen a direct practical application of standard deviation, so I'm not aware of how it might be used on the fly to change one's tactical decisions. I think there's a clear application after the fact--standard deviation is a statistical-evaluation abstraction, after all--in determining the values of individual units and models.
As is done to evaluate sports teams, one could create a "box score" for each unit--tracking stats, such as shots, wounds inflicted and surviving models, from game to game--and then determine the standard deviation of the stats. Units with a small standard deviation would be performing consistently, whether well or poorly, while units with a large standard deviation would be less predictable. That information could then be balanced against point costs, so the player could better understand his army's strength and weakness, and which upgrades are worth taking.
Back to the point, Ckok, I think you have an interesting approach; I'm just not sure I see the practical application. I'd appreciate any further explanation you'd care to give.
Last edited by Holothuria; January 26th, 2009 at 21:22. Reason: Disabling smilies for grinning up my math
First I will answer your questions.
1. All the possible outcomes with a six sided die is known, so the standard deviation is one.
2. I believe that adding or subtracting one from the mean is very easy, if you find that your method is easier thats fine but it is still using a probability distribution method over law of averages.
Holothuria you are right I did a horrible job of explaining how a player can use this tactica on the field I will make another tactica explaining how this one is usefull.
Thanks for your input
this is normally when i come in condeming mathhammer as a blasphemy against the dice gods. yet this time i'm impressed with the level of work put into this, nice.
i'm still not a fan of mathammer of course, prefering to trust my judgement more than some precalculated numbers as to what will and won't work but at the end of the day, its the gamers preogative.
one thing i've always found a bit useless about mathhammer however is the inherient limmitations on timeframe. an average game of warhammer lasts for 6 turns each way, however a mathhammer exercise only ever works in an infinite time frame. the longer you roll the dice the closer to the numbers you will get and so on forth.
still, its a friendlier attempt than most and using solid methodology, deserving of a rep cookie if anything.
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[16:19] <@Alzer> Arky's right though
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Ckok, I think it's that first point that's throwing me. The only way I've ever learned to find standard deviation is a fairly long process:
1. Find the mean of the set
2. Find the deviation of each number in the set (i.e., subtract the mean from each)
3. Square the deviations
4. Average those
5. Find the square root (this, if nothing else, is difficult to do mentally in the middle of a game)
And that's your standard deviation. Obviously, you know some shortcuts I don't, so the utility is more clear to you. I look forward to your further elaboration.
I couldn't help but laugh at this. What is your judgment if not a mental calculation of probability--regardless of time or circumstance? The truth of the matter is that anybody who understands the mechanics of the game can make a reasonable guess as to how well something will work. It's really a small step to formalize that understanding into numbers.
Holothuria you are right again that is the way to find the standard deviation, but I made my assumptions because I was using averages, inorder to do it the correct way I would 1st have to roll dice to get a sample size, and that would have been to much to believe, so I use 1 as the standard deviation because I know it would be close to 1.
Last edited by CKO; January 28th, 2009 at 06:53.
I believe that your math is off. If you take 10 lasgun shots, each with a 50/50 chance of hitting, the mean is indeed 5, but the standard deviation is 3.16 or so. You can check it at this website: Standard Deviation Calculator. Remember to enter all eleven possible results [0-10]
The final result, that there's a 68% chance that the guardsmen will get between 2 and 8 hits, isn't exactly useful, and could have been attained through normal probability.
The problem is, standard deviation is only good for determining the number of successes when the probablility is exactly 50/50. Standard deviation is used when measuring the variance of possible results, but there isn't really a way to add probability into the equation, only the results.
For example, the standard deviation in hits for those guardsmen is exactly the same standard deviation for 10 Orks slugga boyz shooting, which is also the same variance for 10 marines shooting. For standard deviation, the odds of success mean nothing, all that matters are the results. This is because no matter how likely they are to hit, 10 shots always have exactly 11 possible results, from 0 hits to 10 hits. How 'probable' it is to attain hits means nothing when using standard deviation.
That is, unless I'm missing a step in the process somewhere. I'm admittedly not a math major though, so does anyone know if I've missed something here?
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